Permutations & Combinations
Distribution of identical objects
Grade 11
Question:
<p><strong>Statement-1:</strong> The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is empty is \({}^9C_3\).</p><p><strong>Statement-2:</strong> The number of ways of choosing any 3 places from 9 different places is \({}^9C_3\).</p>
<p>Statement-1 is true, Statement-2 is false.</p>
<p>Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.</p>
<p>Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.</p>
<p>Statement-1 is false, Statement-2 is true.</p>
Step-by-Step Solution
Key Concept: Statement-1 uses stars and bars with the constraint that no box is empty (requiring at least 1 ball per box), which transforms the problem into distributing 6 remaining balls into 4 boxes freely—this is equivalent to arranging 3 dividers among 9 positions. Statement-2 is simply choosing 3 items from 9.
<p><strong>Step 1: Analyze Statement-1</strong></p><p>Distribute 10 identical balls in 4 distinct boxes with no box empty. First place 1 ball in each box (uses 4 balls), leaving 6 balls to distribute freely among 4 boxes.</p><p><strong>Step 2: Apply Stars and Bars</strong></p><p>Distributing 6 identical balls in 4 distinct boxes with no restriction uses the formula: $\binom{6+4-1}{4-1} = \binom{9}{3} = {}^9C_3$</p><p><strong>Step 3: Analyze Statement-2</strong></p><p>Choosing 3 places from 9 different places is simply $\binom{9}{3} = {}^9C_3$</p><p><strong>Step 4: Compare Statements</strong></p><p>Both statements are TRUE and give the same answer ${}^9C_3$. However, they arrive at this through different reasoning: Statement-1 through constrained distribution (stars and bars), Statement-2 through direct selection.</p><p><strong>Step 5: Determine Relationship</strong></p><p>Statement-2 does NOT explain why Statement-1 is true. They are independently true. The correct answer depends on whether the question asks if Statement-2 is a correct explanation for Statement-1 (typically the answer would be that Statement-1 is true, Statement-2 is true, but Statement-2 does not explain Statement-1).</p><p>∴ Answer: B</p>
Correct Answer: B